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Mirrors > Home > MPE Home > Th. List > Mathboxes > sblpnf | Structured version Visualization version GIF version |
Description: The infinity ball in the absolute value metric is just the whole space. 𝑆 analogue of blpnf 23009. (Contributed by Steve Rodriguez, 8-Nov-2015.) |
Ref | Expression |
---|---|
sblpnf.s | ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) |
sblpnf.d | ⊢ 𝐷 = ((abs ∘ − ) ↾ (𝑆 × 𝑆)) |
Ref | Expression |
---|---|
sblpnf | ⊢ ((𝜑 ∧ 𝑃 ∈ 𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sblpnf.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ}) | |
2 | elpri 4591 | . . 3 ⊢ (𝑆 ∈ {ℝ, ℂ} → (𝑆 = ℝ ∨ 𝑆 = ℂ)) | |
3 | sblpnf.d | . . . . 5 ⊢ 𝐷 = ((abs ∘ − ) ↾ (𝑆 × 𝑆)) | |
4 | eqid 2823 | . . . . . . 7 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) = ((abs ∘ − ) ↾ (ℝ × ℝ)) | |
5 | 4 | remet 23400 | . . . . . 6 ⊢ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (Met‘ℝ) |
6 | xpeq12 5582 | . . . . . . . . 9 ⊢ ((𝑆 = ℝ ∧ 𝑆 = ℝ) → (𝑆 × 𝑆) = (ℝ × ℝ)) | |
7 | 6 | anidms 569 | . . . . . . . 8 ⊢ (𝑆 = ℝ → (𝑆 × 𝑆) = (ℝ × ℝ)) |
8 | 7 | reseq2d 5855 | . . . . . . 7 ⊢ (𝑆 = ℝ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (ℝ × ℝ))) |
9 | fveq2 6672 | . . . . . . 7 ⊢ (𝑆 = ℝ → (Met‘𝑆) = (Met‘ℝ)) | |
10 | 8, 9 | eleq12d 2909 | . . . . . 6 ⊢ (𝑆 = ℝ → (((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ↔ ((abs ∘ − ) ↾ (ℝ × ℝ)) ∈ (Met‘ℝ))) |
11 | 5, 10 | mpbiri 260 | . . . . 5 ⊢ (𝑆 = ℝ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆)) |
12 | 3, 11 | eqeltrid 2919 | . . . 4 ⊢ (𝑆 = ℝ → 𝐷 ∈ (Met‘𝑆)) |
13 | relco 6099 | . . . . . . . . 9 ⊢ Rel (abs ∘ − ) | |
14 | resdm 5899 | . . . . . . . . 9 ⊢ (Rel (abs ∘ − ) → ((abs ∘ − ) ↾ dom (abs ∘ − )) = (abs ∘ − )) | |
15 | 13, 14 | ax-mp 5 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ dom (abs ∘ − )) = (abs ∘ − ) |
16 | absf 14699 | . . . . . . . . . . . 12 ⊢ abs:ℂ⟶ℝ | |
17 | ax-resscn 10596 | . . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ | |
18 | fss 6529 | . . . . . . . . . . . 12 ⊢ ((abs:ℂ⟶ℝ ∧ ℝ ⊆ ℂ) → abs:ℂ⟶ℂ) | |
19 | 16, 17, 18 | mp2an 690 | . . . . . . . . . . 11 ⊢ abs:ℂ⟶ℂ |
20 | subf 10890 | . . . . . . . . . . 11 ⊢ − :(ℂ × ℂ)⟶ℂ | |
21 | fco 6533 | . . . . . . . . . . 11 ⊢ ((abs:ℂ⟶ℂ ∧ − :(ℂ × ℂ)⟶ℂ) → (abs ∘ − ):(ℂ × ℂ)⟶ℂ) | |
22 | 19, 20, 21 | mp2an 690 | . . . . . . . . . 10 ⊢ (abs ∘ − ):(ℂ × ℂ)⟶ℂ |
23 | 22 | fdmi 6526 | . . . . . . . . 9 ⊢ dom (abs ∘ − ) = (ℂ × ℂ) |
24 | 23 | reseq2i 5852 | . . . . . . . 8 ⊢ ((abs ∘ − ) ↾ dom (abs ∘ − )) = ((abs ∘ − ) ↾ (ℂ × ℂ)) |
25 | 15, 24 | eqtr3i 2848 | . . . . . . 7 ⊢ (abs ∘ − ) = ((abs ∘ − ) ↾ (ℂ × ℂ)) |
26 | cnmet 23382 | . . . . . . 7 ⊢ (abs ∘ − ) ∈ (Met‘ℂ) | |
27 | 25, 26 | eqeltrri 2912 | . . . . . 6 ⊢ ((abs ∘ − ) ↾ (ℂ × ℂ)) ∈ (Met‘ℂ) |
28 | xpeq12 5582 | . . . . . . . . 9 ⊢ ((𝑆 = ℂ ∧ 𝑆 = ℂ) → (𝑆 × 𝑆) = (ℂ × ℂ)) | |
29 | 28 | anidms 569 | . . . . . . . 8 ⊢ (𝑆 = ℂ → (𝑆 × 𝑆) = (ℂ × ℂ)) |
30 | 29 | reseq2d 5855 | . . . . . . 7 ⊢ (𝑆 = ℂ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) = ((abs ∘ − ) ↾ (ℂ × ℂ))) |
31 | fveq2 6672 | . . . . . . 7 ⊢ (𝑆 = ℂ → (Met‘𝑆) = (Met‘ℂ)) | |
32 | 30, 31 | eleq12d 2909 | . . . . . 6 ⊢ (𝑆 = ℂ → (((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆) ↔ ((abs ∘ − ) ↾ (ℂ × ℂ)) ∈ (Met‘ℂ))) |
33 | 27, 32 | mpbiri 260 | . . . . 5 ⊢ (𝑆 = ℂ → ((abs ∘ − ) ↾ (𝑆 × 𝑆)) ∈ (Met‘𝑆)) |
34 | 3, 33 | eqeltrid 2919 | . . . 4 ⊢ (𝑆 = ℂ → 𝐷 ∈ (Met‘𝑆)) |
35 | 12, 34 | jaoi 853 | . . 3 ⊢ ((𝑆 = ℝ ∨ 𝑆 = ℂ) → 𝐷 ∈ (Met‘𝑆)) |
36 | 1, 2, 35 | 3syl 18 | . 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝑆)) |
37 | blpnf 23009 | . 2 ⊢ ((𝐷 ∈ (Met‘𝑆) ∧ 𝑃 ∈ 𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆) | |
38 | 36, 37 | sylan 582 | 1 ⊢ ((𝜑 ∧ 𝑃 ∈ 𝑆) → (𝑃(ball‘𝐷)+∞) = 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 ⊆ wss 3938 {cpr 4571 × cxp 5555 dom cdm 5557 ↾ cres 5559 ∘ ccom 5561 Rel wrel 5562 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 +∞cpnf 10674 − cmin 10872 abscabs 14595 Metcmet 20533 ballcbl 20534 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-xneg 12510 df-xadd 12511 df-xmul 12512 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-psmet 20539 df-xmet 20540 df-met 20541 df-bl 20542 |
This theorem is referenced by: dvconstbi 40673 |
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